Introduction to Acoustics

1128 Part H Engineering Acoustics
Part H 28.2
be expressed as a linear combination of the natural frequencies
and their reciprocals. When these restrictive
proportionality conditions do not exist, the structure may
be better modeled by complex mode shapes.
Complex modes result when the instantaneous velocity
at each DOF is treated as being independent of
the displacement. This doubles the number of system
differential equations, but simplifies them from second
order to first order. The solution vectors are most
commonly written in terms of complex displacements,
one for each DOF. Expressing the modal displacements
in this manner eliminates any constraints between the
damping factors and the natural frequencies. All but
the simplest experimental curve-fitting algorithms can
identify complex modes. Most finite-element codes are
restricted to the undamped analysis of structures and can
therefore only identify the approximating real or normal
modes.
Each modal vector represents a displacement pattern.
A zero in the vector denotes a node, a point and
direction on the structure that does not move in that
mode. The element in a modal vector that is largest
in displacement value is termed an anti-node. A given
mode shape can be readily excited by a sinusoidal force
at or near its natural frequency applied to an anti-node.
The same force applied at a node will impart no motion
in that shape.
It should be noted that the real or complex displacement
values in each modal vector are relative numbers.
The mode shape is inferred by the ratio between the
vector elements, not their specific values. The elements
within each modal vector may be scaled by any of various
methods, the simplest of which is division by the
anti-node value. This is useful for display or plotting
28.2 Experimental Modal Testing
Modal testing is a systematic method for identification
of the modal parameters of a structure. Generally these
include natural frequencies, modal damping, and UMM
mode shapes. In experimental modal testing the structure
is excited with a measured force at one or more
points and the response is determined at one or more
points. From these sets of data, the modal parameters
are determined, often by the use of multidimensional
curve-fitting routines on a digital computer.
Modal testing may use sinusoidal, random, pseudorandom,
or impulsive excitation. The response may be
measured mechanically, optically, or indirectly (by ob-
so that all modes have a common maximum value.
However, the inner product or length of each vector is
related to the modal mass of the mode. For many analytical
purposes, including structural dynamic modification,
it is necessary to choose the length of each vector so that
the corresponding modal mass is equal to 1. When the
shape vectors are scaled in this manner, they are said
to be orthonormal or unit modal mass (UMM) modal
vectors.
It is normally possible to excite a mode of vibration
from any point on a structure that is not a node
and to observe motion at all points that are not nodes.
A mode shape is a characteristic only of the structure itself,
independent of the way it is excited or observed.
In practical terms, however, instrumentation associated
with both excitation and observation may modify the
structure slightly by adding mass or stiffness (or both).
This results in small shifts in frequency and mode shape,
which in most cases are negligible. Acoustic excitation
and optical monitoring are the least intrusive, but mechanical
means of driving and observing are frequently
more convenient and less costly.
Mode shapes are unique for a structure, whereas the
deflection of a structure at a particular frequency, called
an operating deflection shape (ODS), may result from
the excitation of more than one normal mode. When exciting
a structure at a resonance frequency, the ODS will
be determined mainly by one mode, although if several
modes have nearly the same frequency, special techniques
may be required to determine their contributions
to the observed ODS. Modes of a structure are functions
of the entire structure. A mode shape describes how every
point on the structure moves when it is excited at
any point.
serving the radiated sound field, for example). The first
step in experimental modal testing is generally to obtain
a set of frequency response functions.
There is a vast amount of good literature describing
experimental modal testing. Ewins [28.1] provides
a good overall introduction. The proceedings of the annual
International Modal Analysis Conference (IMAC),
held every year since 1982, is a gold mine of papers on
the subject [28.2]. The Structural Dynamics Research
Laboratory at the University of Cincinnati has published
many scientific papers and reports on the subject, and
has included several tutorials on their website [28.3].